Elementary Processes and Rate Theories

• An elementary process cannot be determinedfrom the net stoichiometry of a reaction.

• For example, the combustion of octane is:

2C8H18 + 25O2 −→ 16CO2 + 18H2O

but the reaction does not proceed by 25 oxygenmolecules interacting simultaneously with twooctane molecules.

• Statistics works against this.

• Instead the reaction proceeds by a sequenceof elementary processes involving interac-tions of a small number of species at a time.

• BUT not all stoichiometric reactions that arebimolecular are elementary reactions.

Elementary Processes

• Elementary Processes may be categorized as:

• Unimolecular

• Bimolecular

• Termolecular

• Photochemical

• See table 27.1 on page 1041 of Winn.

Unimolecular Processes

• Isomerization and elimination are examples ofunimolecular reactions.

• The rate of isomerization depends on tempera-ture through the activation energy due to theintramolecular energy barrier.

• If the activation energy is low (as in thecase for conformal isomers such as boat andcrown or staggered and eclipse), the rate ofisomerization is rapid and is not stronglytemperature dependent.

• If the activation energy is high (as in thecase of interconversion of optical isomers)then the rate of isomerization tends to below at room temperature.

• In the case of an elimination reaction, the acti-vation energy is related to the height of the po-tential energy barrier between isolated groundstate reactants and isolated products.

• Excitation of reactants can increase therate of reaction.

Bimolecular Processes

• There are many types of elementary bimolecularprocesses.

• Recombination cannot occur solely by a bi-molecular collision due to conservation of mo-mentum and energy.

• An additional collision is required to carryoff excess energy and momentum.

• Therefore many recombination processesare density dependent.

• Recombination may also occur by radiative as-sociation where a photon is emitted to removethe excess energy.

• This is an example of where the bimolecularprocess has at least two products. In thiscase one of the products is a photon.

• Whether a bimolecular reaction leads to prod-ucts depends on the potential energy surfacewhich describes the energy interaction of all theatoms involved in the system.

• If there are large energy barriers betweenreactants and products then the reactiondoes not occur at an appreciable rate.

Termolecular Reactions

• Pathways not accessible through bimolecular re-actions may be accessed by termolecular reac-tions.

• Termolecular reactions require the simultane-ous or nearly simultaneous interaction of threemolecules.

• The rate of reaction is often strongly pres-sure dependent.

• Consider recombination of I in Ar:

I + I −→ I · · · I

• If I · · · I collision pair can interact with Ar be-fore it fall apart, then energy may be transferredto or from the Ar atom.

• If Ar takes away energy, I · · · I is stabilizedto I2.

• If Ar delivers energy, I · · · I is further desta-bilized.

• Because termolecular reactions require the si-multaneous or nearly simultaneous interactionof three molecules, the rate may increase as thespeed of the interacting species is slowed down.

• Termolecular reactions may be faster atlower temperatures.

• Consider a termolecular reaction as occurring intwo stages:

• transient association:

A+Ak2⇀↽k−1

A · · ·A

• collisional stabilization:

A · · ·A+Mk′2→A2 +M

• If the second process occurs immediatelyupon the first occurring, the interaction ofthree bodies may be regarded as simulta-neous.

• The chaperone mechanism is a variant of this:

• chaperone complex foundation:

A+Mk2⇀↽k−1

A · · ·M

• bimolecular atom transfer:

A · · ·M + Ak′2→A2 +M

• Both mechanisms give the same rate law.

• Which one is preferred depends on the sta-bility of the complex formed.

• The more stable the complex formed in thefirst step, the more likely the second stepwill occur.

Photochemical Processes

• Light must be absorbed to cause a chemicalchange.

• Each photon absorbed leads to one photo-chemically active molecule.

• The rate of reaction depends on the availabilityof photons:

HI + hν → H + I

• One mole of photons is an einstein.

• However the intensity of the light and theduration of irradiation must be considered.

• The Beer-Lambert Law is:

Ia = I0(1− e−abc)

where:

• a is the absorbance coefficient

• b is the optical path length of the sample

• c is the concentration of the absorbingspecies.

• The product abc is dimensionless and there area number of choices.

• If c is in units of molecules per unit volume,b is in units of distance, then a has units ofarea and is known as the absorbance crosssection.

• Not every photon absorbed leads to the photo-chemical product of interest.

• Photochemical product quantum yield Φ is:

Φ =number of molecules produced

number of photons absorbed

• If competing processes may occur, thenΦ < 1.

• Photochemical processes include:

• Excitation of the molecule vibrationally, ro-tationally, and/or electronically.

• Dissociation

• Ionization

• Isomerization

• Energy from absorbed photons may also be lostphotophysically.

• Fluorescence

• Phosphorescence

• Collisional Quenching

Simple Collision Theories of Reactions

• Can be used to predict rate coefficients for sim-ple bimolecular processes.

• Must take into account

• Collision or encounter rate

• Probability of reactive collision.

• Collision or encounter rate determined from:

• Relative collision velocity

• Collision cross section

• Number densities of colliding species

• Probability of a reactive collision takes into ac-count:

• Energy requirements

• Quantum mechanical (spin) requirements

• Collision geometry requirements (steric fac-tor).

Gas Phase Reactions

• Consider the associative detachment reaction:

H + H− → H2 + e−

• Well-studied both experimentally and the-oretically.

• Observed rate coefficient at 300 K is 1.17×1012 M−1 s−1.

• Compare the rate coefficient with the collisionrate.

• Recall that:

Z12 =< gσ > nHnH−

therefore

k = NA < gσ > .

• and recall that:

< g >=

(8kBT

πµ

)1/2

• The hard-sphere cross section is:

σ12 = π(r1 + r2)2

• Assuming that the radii of each of H andH− are approximated by the bohr radius a0

of 5.3 × 10−11 m gives:

σ12 = 3.5× 10−20 m2

• This gives a bimolecular collision rate of

k =Z12

nHnH−= 7.5× 1010 M s−1

at 300 K.

• Why is this less than the observed rate?

• Is the hard-sphere cross section a reasonable es-timate?

• What value of the collision cross sectionwould be consistent with the observed ratecoefficient?

k

< g >= 5.5× 10−19 m2

• How do you explain this?

• Is the bohr radius a reasonable estimate ofthe radius of H−?

• The rate coefficient is observed to be nearly in-dependent of temperature.

• No activation energy is apparent.

• This is typical of ion-molecule reactions.

• Consider a reaction that does have an activationenergy:

Cl + H2 → HCl + H

• Observed rate coefficient at 300 K is 9.4×106 M−1 s−1.

• Observed Arrhenius activation energy is∼ 18.3 kJ mol−1.

• From the observed rate coefficient, a hard-sphere cross section is estimated to be8.6 × 10−24 m2, which is absurdly small.

• Probability of reaction must be considered.

• Fraction of collisions with energy in excessof cross section is 6.5 ×10−4.

• This appears to be sufficient to account forthe difference between the collisional rateand the reaction rate.

• For other reactions, this is not sufficient.

• Directness of collision may matter.

• Usually accounted for by the “steric fac-tor”.

Solution Reactions

• Differs from gas phase reactions in that reac-tants must diffuse toward each other.

• Concepts from gas phase such as activation en-ergy is still valid.

• In solution, molecules are always interactingwith the solvent molecules.

• As reactant molecules diffuse toward each other,the molecules of solvent can form a “cage”around them.

• This means that the reactant molecules willbe in proximity for a long time.

• Energy can be rearranged within the reac-tants and gained or lost from the solventmolecules.

• Reactions in solution can fall into two classes inaccordance with the activation energy.

• Reactions with low activation energy take placein the diffusion limited regime.

• The rate of reaction is controlled by therate of diffusion of reactants toward eachother.

• Reactions with high activation energy take placein the activation limited regime.

• Reaction depends on fluctuations withinthe solvent cage that are occasionally largeenough to provide the activation energy.

• The reaction sequence may be representedas:

A + B ⇀↽ A· · ·Bwhich represents reactant diffusion in theforward direction and dissociation of theencounter pair in the reverse direction and:

A· · ·B → C + Dwhich represents reaction of the encounterpair.

• Applying steady state analysis to A· · ·Bleads to the overall rate law:

< =d[C]

dt= k′1[A · · ·B] =

k2k′1

k−1 + k′1[A][B]

with

kexp =k2k′1

k−1 + k′1

• This expression allows us to assess the relativeroles of diffusion and dissociation of theencounter pair,

• In the diffusion controlled regime,

k′1 >> k−1 and kexp = k2

• In the activation limited regime,

k′1 << k−1 and kexp =k2k′1

k−1= k′1Keq

where Keq is for the diffusive formation ofthe encounter pair.

• Keq is usually small because:

• The entropy of formation of the encounterpair is negative.

• The enthalpy of formation of the encounterpair is positive or slightly negative.

• Thus ∆rG◦

is usually positive.

• The diffusion rate coefficient k2 may be esti-mated from the diffusion transport equation.

• The diffusion coefficients DA and DB are im-portant.

• These are functions of temperature, the vis-cosity of the solvent, and molecular radiisince:

D =kBT

6πηR

• If either A or B are ions, then electrostatic forcesplay a role.

• Also to be considered is detailed balance.

• Detailed balance uses concentrations, notactivities.

• Connected to thermodynamic equilibriumby fugacity, activity, and other correctionsfor nonideality.

• Initially consider uncharged reactants.

• A is a spherical molecule with characteristicradius RA.

• Spherical molecules of B diffuse around Ain a structureless continuous solvent.

• An encounter complex A· · ·B forms whenA and B are separated by R∗.

• Because B is now part of A· · ·B, [B] = 0 at R∗.

• This gives a radial concentration gradient whichdrives the diffusion of B towards A.

• This described by Fick’s First Law:

JB,r = −DB∂nb∂r

= −DBd[B]

drNA(1000 L m

−3)

where:

• JB,r is the radial flux,

• DB is the diffusion coefficient for B in thatsolvent system

• nB is B number density

• NA and (1000 L m−3) convert from con-centration in moles L−1 to number densityin molec m−3.

• This may be diffentiated to give an expressionfor ∂2nB/∂r

2 which can be substituted intoFick’s Second Law:

∂nB(r, t)

∂t= D

∂2nB∂r2

• Thus:∂nB(r, t)

∂t= −∂JB

∂r

• But if there is a steady state, ∂nB/∂t = 0,then ∂JB/∂r = 0 and the flux is the samethrough a spherical shell at any distance rabout A.

• Total radial flow through a sphere of radiusr is the radial flux times the surface area4πr2

• Thus the inward flow is:

Ir = −4πr2JB,r = 4πr2DBNA(1000 L m−3)d[B]

dr)

• Because the number of B molecules is conserved,Ir must be a constant and independent of r.Therefore:

r2 d[B]

dr=

Ir

4πDBNA(1000 L m−3

)

which has the solution:

[B] = [B]∞ −Ir

4πrDBNA(1000 L m−3)

where [B]∞ is the bulk concentration of B inthe large distance limit.

• Since [B] = 0 at r = R∗,

[B]∞ =Ir

4πR∗DBNA(1000 L m−3)

or

Ir = [B]∞4πR∗DBNA(1000 L m−3)

• Thus for r > R∗, the concentration profile for[B] becomes:

[B] = [B]∞

(1− R∗

r

)

• As indicated in figure 27.3, the gradient can bequite long range.

• Recall that all of the above is with respect to Bdiffusing toward A.

• Now consider the diffusion of A and B towardeach other:

• This entails replacing DB with DA +DB.

• The total radial flow is the rate of formation ofA · · ·B per A molecule.

• Thus the reaction rate is:

< =d[A · · ·B]

dt= −d[B]

dt= k2[A][B]

• Recalling that [B] in the above refers the bulkconcentration, called [B]∞ in the derivation ofradial diffusion, the rate becomes:

< = [A]Ir

• With the correct diffusion coefficient, thisbecomes:

< = k2[A][B]

=[4πR∗(DA +DB)NA(1000 L m

−3)]

[A][B]

and the diffusion rate coefficient k2 can beidentified:

k2 =[4πR∗(DA +DB)NA(1000 L m−3)

]

• Thus a maximum rate of reaction is solutionin the diffusion controlled regime may be esti-mated.

• Assuming that R∗ = 5 A and DA +DB =10−9 m2 s−1 yields a rate coefficient k2 =4× 109 M−1 s−1.

• When this is compared with the gas phase bi-molecular rate coefficient, it may be noted that:

• The gas phase rate coefficient is propor-tional to the relative speed times a crosssection.

• The solution rate coefficient is proportionalto a diffusion coefficient times a length.

• The gas phase rate coefficient is propor-tional to the square of a characteristic re-action radius.

• The solution rate coefficient appears as alinear function of a characteristic reactionradius.

• But the diffusion coefficient depends on vis-cosity and molecular size.

• The dependence of the solution rate coefficienton molecular size can be further explored by theconsideration of viscosity.

• Recall the Stokes-Einstein relation:

D =kBT

6πηR

• Applying this to DA +DB gives:

DA +DB =kBT

6πη

(1

RA+

1

RB

)

• If it is assumed that R∗ = RA + RB, thesum of the hydrodynamic radii of A and B,then

k2 =2kBT

3η

(RA +RB)2

RARBNA(1000 L m

−3)

• If it is further assumed that RA = RB andnoting that the universal gas constant isR = kNA then:

k2 =8RT

3η(1000 L m

−3)

• Thus k2 is independent of the size of themolecule.

• Two effects cancel each other out.

• Large molecules diffuse more slowly.

• Large molecules present a larger target.

• A value for k2 may be estimated based on vis-cosity.

• Water has a viscosity of 10−3 Pa s at 300K.

• This predicts k2 = 7×109 M−1 s−1 which isin reasonable agreement with the previousestimate.

• The viscosity dependence of diffusion con-trolled rate coefficients has been confirmedby studies of mixed solvent systems of vary-ing viscosity.

• The above expressions apply to neutral-neutralinteractions and were first derived by Smolu-chowski in the context of colloidal particlegrowth.

• Additional factors must be considered in thecase of ionic reactants.

• Electrostatic interactions, whether ion-ionor ion-neutral, are longer ranged thanneutral-neutral interactions.

• Ion-ion interactions of like charge will diffusemore slowly than neutral-neutral.

• Ion-ion interactions of unlike charge will diffusemore quickly than neutral-neutral.

• Therefore, Fick’s First Law needs to bemodified to account for the electrostatic poten-tial energy gradient in addition to the concen-tration gradient, since the chemical potentialgradients will be a function of the two gradi-ents.

• Thus the rate coefficient becomes:

k2 = 4πR∗(DA +DB)NA(1000 L m−3

)f

where f is the electrostatic factor:

f =R0

R∗(eR0/R∗ − 1)

and R0 is the distance at which the electrostaticenergy is kBT , the characteristic thermal en-ergy.

• Recall that the Colomb energy is:

E =zAzBe

2

4πε0εrR

where

• zA and zB are the charge numbers of A andB

• ε0 is the permittivity of free space

• εr is the relative permittivity of the solvent.

• Since this energy is equal to kBT at R0,

R0 =zAzBe

2

4πε0εrkBT

• For water at 25◦C:

• εr = 78.3

• R0 = zAzB(7.16× 10−10) m.

Thus at R∗ = 5 A:

• f = 1.88 for zAzB = -1.

• f = 0.45 for zAzB = 1.

• f = 5.7 for zAzB = -4.

• f = 0.019 for zAzB = 4.

• This does not take into account the possibilityof other ions affecting A and B as they diffusetoward each other.

• If the total ion concentration is 10−4 M orgreater, then corrections must be appliedto account for the primary salt effect.

• The Debye-Huckel expression for ionic ac-tivity coefficients are useful.

The Transition State

• Consider the potential energy surface for the re-action

• in particular, the forces among the atomsat various geometries along the way fromreactants to products.

• Rigourously, one should consider all spatialcoordinates

• In practice only a few geometric variableschange as the reaction proceeds.

• Consider a case with only three spatial vari-ables, the H + H2 → H2 + H reaction.

• May be studied through isotopic substitu-tion, D + H2 → HD + H

• Only three geometric parameters need tobe considered, either three atom-atom sep-arations or two atom-atom separations andan angle.

• If one of these is fixed, then there are onlytwo coordinates required.

• If the angle is fixed, then the distances fromone atom to each of the other atoms are allthat is required to completely describe thesystem.

• The full dimensional potential energy surface forH + H2 is very well known.

• In the ground electronic state, H3 is not astable electronic species.

• In the course of the exchange reaction, thethree atoms will be close together.

• The forces are such as to push the atomstoward a linear conformation.

• The transition state of a reaction is ill-definedin terms of configuration.

• Represents an arrangement of atoms that isneither isolated reactants nor isolated prod-ucts.

• The concept of transition state is widelyused in interpreting reactions.

• Accurate chemical potentials may bedetermined quantum mechanically for systemswith few electrons.

• It is difficult to perform accurate calcula-tions for many electron systems.

• Therefore approximate methods are used.

• These approximate methods includes.

• LEPS (London, Eyring, Polanyi, and Sato,1928) combines diatomic Morse potentialswith adjustable “interaction terms” whichmatch empirical energies.

• The existence of a potential energy barrier alongthe path from reactants to product suggests themicroscopic origin of activation energy.

• Barrier height is not equal to the activationenergy.

• The classical reaction coordinate may be definedby the minimum energy path between reactantsand products.

• This path will go through the saddle point,if it exists for that potential.

• It is classical in that it does not take intoaccount quantum effects such as tunnellingor zero point energy.

• Consider the saddle point, which is sometimesidentified as the transition state.

• A saddle point, by definition, is a maxi-mum in one direction and a minimum inthe other.

• It sits on the classical reaction coordinate.

• Motion along q1, a line tangential to theclassical reaction coordinate correspondsassymmetric stretch ( R1 + R2 = constant).

• Motion perpendicular to the reaction coor-dinate along q2 (i.e. along the R1 = R2

line) corresponds to symmetric stretch.

• Motion in the q1 direction is not bound.

• Because the potential has a maximumalong q1, the second derivative is negative.

∂2U

∂q21

< 0

• This means that the correspondingharmonic force constant is negative, whichin turn corresponds to a negative force con-stant.

• This means that the harmonic oscillator (ornormal mode) frequency is an imaginarynumber.

ωi =

√normal mode force constant

normal mode reduced mass

• Note that the potential energy surface is specificto the atoms and the electronic state involved.

• For example the H+ + H2 → H2 + H+

potential has a local minimum when thethree atoms are close together.

• The energy of the “transition state” is atthe local minimum when the atoms are ina triangular configuration.

• The existence of a local minimum is associatedwith a potentially stable intermediate that canexist as a distinct species.

• Examples include O + CO→ CO + O (sta-ble CO2), O + N2 + ON + N (stable N2O).

• Potential energy surfaces for different electronicconfigurations of the same atoms are not alwayswell separated.

• Each potential energy surface will have itsown transition state and reaction coordi-nate.

• Surfaces can intersect and a reaction cancross intersections of the potentials.

• It is the intersections of these potential en-ergy surfaces that gives rise to branching ofchemical reactions.

• Minima associated with transition states can-not be the absolute minimum of the potentialenergy surface.

• If it were, it would be a stable compoundand would be classified as an intermediaterather than a transition state.

Activated Complex Theory

• A transition state is often identified as an acti-vated complex.

• In 1935, Eyring and, independently Evans andPolanyi, published theory about bimolecularrate coefficients.

• This theory is known as:

• Transition state theory

• Absolute rate theory

• Activated complex theory

• This theory considers

• the reaction potential energy surface

• the transition state

• the thermal distribution of collision ener-gies.

• statistical mechanics and partitionfunctions

• Results have

• reasonable accuracy

• important physical insights

• opportunities for improvement

• Consider activated complex theory (ACT) inthe context of the exchange reaction A + BC→ AB + C.

• The activated complex is ABC†.• The rate coefficient is k2 at temperature T.

• Assume classical mechanics describes thereaction.

• Quantum effects will be considered later.

• The assumption of constant T allows the use ofequilibrium distribution functions for reactantsand activated complex.

• Assume that every activated complex goes toproducts.

• This violates microscopic reversibility sothis will have to be reexamined later.

• Start with the system’s Hamiltonian, H, whichdescribes the total kinetic and potential energyof the system.

• The potential energy may be considered interms of the intermolecular potential forthe system and the intramolecular poten-tial within the molecules.

• The Hamiltonian is written in terms of co-ordinates qi and momenta pi.

• The potential energy depends on the coor-dinates qi.

• The kinetic energy depends on themomenta pi.

• For a three atom system there will be 18variables, 9 momenta pi and 9 coordinatesqi.

• Imagine the associated 18 dimensional cartesianspace:

• Coordinates and momenta selected to beorthogonal

• This 18 dimensional cartesian space isknown as phase space.

• A volume element, dτ , in phase space is 18dimensional:

dτ = dq1 . . . dq9dp1 . . . dp9

• Generally for N particles, the associatedphase space is 6N dimensional.

• The reaction coordinate is denoted q1.

• A surface perpendicular to this coordinateis constructed in phase space.

• This surface is known as a dividing surface.

• At this point your text, like many othertreatments of transition state theory, stopsreferring to the momenta and continues torefer to coordinates.

• All coordinates far to one side of the divid-ing surface are clearly reactants.

• All coordinates far to the other side of thedividing surface are clearly products.

• The dividing surface is through the transi-tion state on the potential.

• Consider the fraction of the total of N systemsin a volume element dτ :

d18N

N=

(Boltzmann Factor)(Volume Element)

Partition Function

=e−βHdτ/h9

∫e−βHdτ/h9

where β = 1/kT .

• This is exact at equilibrium.

• It is assumed that the reacting system doesnot deviate significantly from equilibrium.

• h9 may be thought of as a scaling factor forthe volume element since dqidpi has unitsof kg m2 s−1 or J s.

• The expression is now integrated over the pos-itive values of pi to find the rate at which sys-tems cross the dividing surface from reactantsto products.

• When integrating the denominator, takenote that most of the time A is far from BCand thus their interaction potential may beignored. Thus the demoninator may betreated as the product of the partition func-tions for A and for BC.

• When integrating the numerator, integrateover all the coordinates and momenta, ex-cept the reaction coordinate.

• The numerator is in the form of a partition

function for the activated complex, ABC†,

corrected for the fact that the internal en-ergy of ABC† is measured from the energyat the barrier height.

• This yields:

where q† is the partition function for the acti-vated complex with the reaction coordinate re-moved.

• This represents the rate at which reactantsapproach the dividing surface.

• Since it is assumed that all systems whichreach the dividing surface go on to prod-ucts, this leads directly to the rate of reac-tion:

< = k2[A][BC] = d[ABC†]dt

=dN

dt

1

V NA(1000 L m−3)

where V is the volume of the system.

• Must now consider exactly what N means inthese equations.

• In this context, let the number of A atomsbe represented by NA and the number of

BC molecules be represented by NBC .

d18N

N=d6NANA

d12NBCNBC

Thus N = NANBC and is the number ofABC complexes possible.

• Expressing N’s in terms of concentration:

< = k2[A][BC] =

[A][BC]kBT

h

(q†

qAqBCe−U†/kBT

)V NA(103 L m−3)

where NA now represents Avogadro’s number.

• Thus the rate coefficient is:

k2 =kBT

h

(q†

qAqBCe−U†/kBT

)V NA(103 L m−3)